# American Institute of Mathematical Sciences

2013, 33(8): 3517-3541. doi: 10.3934/dcds.2013.33.3517

## On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations

 1 Department of Mathematics, Zhejiang University, Hangzhou 310027 2 Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

Received  September 2012 Revised  November 2012 Published  January 2013

This paper is devoted to the study of the inhomogeneous hyperdissipative Navier-Stokes equations on the whole space $\mathbb{R}^N,N\geq3$. Compared with the classical inhomogeneous Navier-Stokes, the dissipative term $- Δ u$ here is replaced by $D^2u$, where $D$ is a Fourier multiplier whose symbol is $m(\xi)=|\xi|^{\frac{N+2}{4}}$. For arbitrary small positive constants $ε$ and $δ$, global well-posedness is showed for the data $(\rho_0, u_0)$ such that $(ρ_{0} - 1, u_0)∈ H^{\frac{N}{2}+ε} × H^{δ}$ with $\inf_{x\in \mathbb{R}^N}\rho_0>0$. To our best knowledge, this is the first result on the inhomogeneous hyperdissipative Navier-Stokes equations, and it can also be viewed as the high-dimensional generalization of the 2D result for classical inhomogeneous Navier-Stokes equations given by Danchin [Local and global well-posedness results for flows of inhomogeneous viscous fluids. Adv. Differential Equations 9 (2004), 353--386.]
Citation: Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3517-3541. doi: 10.3934/dcds.2013.33.3517
##### References:
 [1] H. Abidi, G. Gui and P. Zhang, On the decay and stability of global solutions to the 3D inhomogeneous Navier-Stokes equations,, Comm. Pure Appl. Math., 64 (2011), 832. doi: 10.1002/cpa.20351. [2] H. Abidi, G. Gui and P. Zhang, On the Wellposedness of three-dimensional inhomogeneous Navier-Stokes equations in the critical spaces,, Arch. Rational Mech. Anal., 204 (2012), 189. doi: 10.1007/s00205-011-0473-4. [3] S.-N. Antontsev, A.-V. Kazhikhov and V.-N. Monakhov, "Boundary Value Problems in Mechanics of Nonhomogeneous Fluids,'' Translated from the Russian,, Studies in Mathematics and its Applications, 22 (1990). [4] H. Bahouri, J.-Y. Chemin and R. Danchin, "Fourier Analysis and Nonlinear Partial Differential Equations,'', Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343 (2011). doi: 10.1007/978-3-642-16830-7. [5] J.-M. Bony, Calcul symbolique et propagation des singularités pour équations aux dérivées partielles non linéaires,, Annales Scinentifiques de l'École Normale Supérieure, 14 (1981), 209. [6] J.-Y. Chemin, Localization in Fourier space and Navier-Stokes system, in "Phase Space Analysis of Partial Differential Equations,'', Vol. I, (2004), 53. [7] J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes,, Journal of Differential Equations, 121 (1995), 314. doi: 10.1006/jdeq.1995.1131. [8] R. Danchin, Density-dependent incompressible viscous fluids in critical spaces,, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1311. doi: 10.1017/S030821050000295X. [9] R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids,, Adv. Differential Equations, 9 (2004), 353. [10] R. Danchin, The inviscid limit for density-dependent incompressible fluids,, Ann. Fac. Sci. Toulouse Math., 15 (2006), 637. [11] R. Danchin, Uniform estimates for transport-diffusion equations,, J. Hyperbolic Differ. Equ., 4 (2007), 1. doi: 10.1142/S021989160700101X. [12] B. Desjardins, Global existence results for the incompressible density-dependent Navier-Stokes equations in the whole space,, Differential Integral Equations, 10 (1997), 587. [13] B. Desjardins, Linear transport equations with initial values in Sobolev spaces and application to the Navier-Stokes equations,, Differential Integral Equations, 10 (1997), 577. [14] B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids,, Arch. Rational Mech. Anal., 137 (1997), 135. [15] R. J. DiPerna and P.-L. Lions, Ordinary differential equations,transport theory and Sobolev spaces,, Invent. Math., 98 (1989), 511. doi: 10.1007/BF01393835. [16] H. Fujita and T. Kato, On the Navier-Stokes initial value problems. I,, Arch. Rational Mech. Anal., 16 (1964), 269. [17] G. Gui, J. Huang and P. Zhang, Large global solutions to 3-D inhomogeneous Navier-Stokes equations slowly varying in one variable,, J. Funct. Anal., 261 (2011), 3181. doi: 10.1016/j.jfa.2011.07.026. [18] G. Gui and P. Zhang, Global smooth solutions to the 2-D inhomogeneous Navier-Stokes equations with variable viscosity,, Chin. Ann. Math. Ser. B, 30 (2009), 607. doi: 10.1007/s11401-009-0027-3. [19] S. Itoh and A. Tani, Solvability of nonstationary problems for nonhomogeneous incompressible fluids and the convergence with vanishing viscosity,, Tokyo Joural of Mathematics, 22 (1999), 17. doi: 10.3836/tjm/1270041610. [20] N. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation,, Geom. Funct. Anal., 12 (2002), 355. doi: 10.1007/s00039-002-8250-z. [21] O. Ladyzhenskaja and V. Solonnikov, The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids,, Journal of Soviet Mathematics, 9 (1978), 697. [22] P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models,'', Oxford Lecture Series in Mathematics and its Applications, 3 (1996). [23] T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation,, Anal. PDE, 2 (2009), 361. doi: 10.2140/apde.2009.2.361.

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##### References:
 [1] H. Abidi, G. Gui and P. Zhang, On the decay and stability of global solutions to the 3D inhomogeneous Navier-Stokes equations,, Comm. Pure Appl. Math., 64 (2011), 832. doi: 10.1002/cpa.20351. [2] H. Abidi, G. Gui and P. Zhang, On the Wellposedness of three-dimensional inhomogeneous Navier-Stokes equations in the critical spaces,, Arch. Rational Mech. Anal., 204 (2012), 189. doi: 10.1007/s00205-011-0473-4. [3] S.-N. Antontsev, A.-V. Kazhikhov and V.-N. Monakhov, "Boundary Value Problems in Mechanics of Nonhomogeneous Fluids,'' Translated from the Russian,, Studies in Mathematics and its Applications, 22 (1990). [4] H. Bahouri, J.-Y. Chemin and R. Danchin, "Fourier Analysis and Nonlinear Partial Differential Equations,'', Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343 (2011). doi: 10.1007/978-3-642-16830-7. [5] J.-M. Bony, Calcul symbolique et propagation des singularités pour équations aux dérivées partielles non linéaires,, Annales Scinentifiques de l'École Normale Supérieure, 14 (1981), 209. [6] J.-Y. Chemin, Localization in Fourier space and Navier-Stokes system, in "Phase Space Analysis of Partial Differential Equations,'', Vol. I, (2004), 53. [7] J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes,, Journal of Differential Equations, 121 (1995), 314. doi: 10.1006/jdeq.1995.1131. [8] R. Danchin, Density-dependent incompressible viscous fluids in critical spaces,, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1311. doi: 10.1017/S030821050000295X. [9] R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids,, Adv. Differential Equations, 9 (2004), 353. [10] R. Danchin, The inviscid limit for density-dependent incompressible fluids,, Ann. Fac. Sci. Toulouse Math., 15 (2006), 637. [11] R. Danchin, Uniform estimates for transport-diffusion equations,, J. Hyperbolic Differ. Equ., 4 (2007), 1. doi: 10.1142/S021989160700101X. [12] B. Desjardins, Global existence results for the incompressible density-dependent Navier-Stokes equations in the whole space,, Differential Integral Equations, 10 (1997), 587. [13] B. Desjardins, Linear transport equations with initial values in Sobolev spaces and application to the Navier-Stokes equations,, Differential Integral Equations, 10 (1997), 577. [14] B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids,, Arch. Rational Mech. Anal., 137 (1997), 135. [15] R. J. DiPerna and P.-L. Lions, Ordinary differential equations,transport theory and Sobolev spaces,, Invent. Math., 98 (1989), 511. doi: 10.1007/BF01393835. [16] H. Fujita and T. Kato, On the Navier-Stokes initial value problems. I,, Arch. Rational Mech. Anal., 16 (1964), 269. [17] G. Gui, J. Huang and P. Zhang, Large global solutions to 3-D inhomogeneous Navier-Stokes equations slowly varying in one variable,, J. Funct. Anal., 261 (2011), 3181. doi: 10.1016/j.jfa.2011.07.026. [18] G. Gui and P. Zhang, Global smooth solutions to the 2-D inhomogeneous Navier-Stokes equations with variable viscosity,, Chin. Ann. Math. Ser. B, 30 (2009), 607. doi: 10.1007/s11401-009-0027-3. [19] S. Itoh and A. Tani, Solvability of nonstationary problems for nonhomogeneous incompressible fluids and the convergence with vanishing viscosity,, Tokyo Joural of Mathematics, 22 (1999), 17. doi: 10.3836/tjm/1270041610. [20] N. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation,, Geom. Funct. Anal., 12 (2002), 355. doi: 10.1007/s00039-002-8250-z. [21] O. Ladyzhenskaja and V. Solonnikov, The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids,, Journal of Soviet Mathematics, 9 (1978), 697. [22] P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models,'', Oxford Lecture Series in Mathematics and its Applications, 3 (1996). [23] T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation,, Anal. PDE, 2 (2009), 361. doi: 10.2140/apde.2009.2.361.
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